Let \(\mathbf {y}= (y_0,\ldots ,y_k)\)  be a choice of Hamiltonian chords corresponding to a Lagrangian labelling \(\mathbf {L}\) , \(A_0 \in \pi _2^\mathrm {num}(\mathbf {y})\) , \(\mathsf {t}:\lbrace 1,\ldots ,\ell \rbrace  \rightarrow _{\ge 0}^Q\)  a choice of tangency data, and \(\lbrace \Gamma _i\rbrace _{i \in I}\)  a set of combinatorial types of bubble trees as above. Let \(k+2\ell = N\) . Suppose we have fixed \(\mathbf {P}_{k^{\prime },\ell ^{\prime },\mathbf {L}^{\prime }}\)  for all \(2 \le k^{\prime }+2\ell ^{\prime } \le N\)  and all Lagrangian labellings \(\mathbf {L}^{\prime }\) ; and suppose that these satisfy the conditions required of a universal choice of perturbation data.
Then the set of \(\mathbf {P}_{k,\ell ,\mathbf {L}}\)  satisfying those conditions, and such that \(\mathcal {M}^*_{\lbrace \Gamma _i\rbrace }(\mathbf {y},A,\mathsf {t},\mathbf {P})\)  is regular, is comeagre.


The argument is essentially standard.
In the absence of higher tangency conditions (i.e., if \(\mathsf {t}(i) \le 1\)  for all \(i\) ), it is a combination of [1]} with [2]} (modified as in Section  to account for the divisor \(V\) ). The necessary techniques to deal with higher tangency conditions were developed in [3]}, and implemented in a setting very similar to ours in [4]} and [5]}.
The only slightly non-standard aspect of our setup is our requirement that the perturbation data be chosen to be \(\mathrm {Sym}(\ell )\) -equivariant.
